CN103460252A - A method for reconstructing a 3d surface from a 2d sketch - Google Patents

Abstract

The present invention relates to a method for reconstructing a three-dimensional surface in a domain wherein the three-dimensional surface is reconstructed from a two-dimensional sketch comprising at least one stroke with a vector field assigned to the at least one stroke. The method comprises the steps of: interpolating the vector field to form a dense vector field; and reconstructing the surface using the dense vector field; wherein interpolating the vector field comprises minimizing a non-linear functional of the vector field with a curl-free constraint on the dense vector field, the non-linear functional of the vector field involving terms defined over the domain and a term defined over the at least one stroke, and wherein the non-linear functional of the vector field is minimized using an augmented Lagrangian method.

Description

Rebuild the method on 3D surface from the 2D sketch

Technical field

The present invention relates to a kind of method from two dimension (2D) sketch (sketch) reconstruction of three-dimensional (3D) surface.The method can be used on the surface model building system based on sketch for various application.

Background technology

Although can use current available modeling (such as, Maya[1], Auto CAD[2], SolidWorks[3] and CATIA[4]) build and/or operate geometric model, but these current available modelings adopt WIMP(window, icon, menu, pointer usually) the interface example is in order to form and/or operation 3D model, this just requires the user to select operation from menu and floating panel, input parameter in dialog box, and mobile reference mark [5].This modeling process is not directly perceived, thereby takes time and effort, if especially under required 3D model complicated situation.On the other hand, draw (sketching) be a kind of natural, easily and the mode rapped off efficiently.By drawing, complicated shape also can several strokes form.Therefore, in microcomputer modelling, usually with drawing, form the design of early stage prototype stage, then change the design of describing into the 3D model again.

Owing to drawing than being easier to, current research direction, the interface modeling (sketch-based interfaces for modeling, SBIM) be called as based on sketch is intended to robotization and assists the transfer process of sketch to 3D.Due to perception efficiency and the intuitive of SBIM system, researchers explore the SBIM system.These systems [5] have comprised from early stage system (such as, SKETCH[12] and Teddy[13]) to nearest system (such as, SmoothSketch[14] and FiberMesh[15]).Because lack illumination or light and shade clue (such as, the clue in photometric stereo [18], [19] and [20] or the clue from shape from shading [21], [22]), the task of rebuilding the 3D model from the 2D line chart has more challenge than the reconstruction of the 3D based on image.Although based on profile [13], hiding profile [14] and structuring annotation [23], attempt processing this challenge, existing SBIM system still can only be moulded interior among a small circle low complex degree object.These objects are the form of shape existence on a small quantity only, and only can comprise coarse structure or smooth structure or circular configuration.And, in existing SBIM system, seldom consider surface details, such as, the folding line structure.Although made efforts and comprised surface details in existing SBIM system, these effort great majority are unsuccessful.At present, if need surface details, just must add on the object of moulding by for example surperficial amplification technique [24].

Many resurfacing algorithms [25], [26], [27] and [28] have strengthened for generate the integrability condition of correct surface elevation from the degree of tilt field.For the single view modeling, method in list of references [9] is used the constrained optimization with multiple geometrical constraint type, yet single view modeling [10] is used closed duplicate method, so that the apparent contour based in parameter space, inflation constraint and conventional specification are rebuild the 3D curved surface.Yet, [9] method in needs a large number of users interactive in order to provide enough constraints (as the author in list of references [7] is showed, the method in list of references [7], [9] and [10] is being processed sparse degree of tilt, preserved the acutance feature and prevent from there will be problem aspect surface deformation [16]) by moulding required 3D surface.

Be good at distribution local surfaces normal to specify local shape [29] based on observing people, the resurfacing based on stroke completes in two steps in list of references [6] and [7].The first step obtains fine and close surface normal along particular stroke by the linear vector insertion method from sparse user's input vector.Second step, integrated to rebuild height map to fine and close surface normal.At LUMO[6] in, insert vector based on telegraph equation (damped wave equation), yet at ShapePalettes[7] in, used energy functional to minimize, and the main mechanism inserted is based on the parabolic type partial differential equation (PDE) with quadravalence item relevant to surface curvature.In LUMO and ShapePalettes method, governing equation is linear PDE, and therefore, vector inserts can apply the numerical value solver, such as, multiple grid method or fast Fourier transform method (FFT).Yet LUMO and ShapePalettes method all are based on Laplace operator, this operator can strengthen the smoothness of the vector field of inserting.This makes distributed vector be diffused into whole territory, then finally in the end steady state (SS) obtain on average, thereby only generate the smooth reconstruct surface do not there is surface details.

Summary of the invention

The present invention aims to provide a kind of new for rebuild the process useful on 3D surface from the 2D sketch.

Briefly say, the present invention proposes and a kind ofly insert for the non-linear vector of rebuilding the 3D surface, and further proposed the numerical value solver inserted for this non-linear vector.

A first aspect of the present invention is a kind of method for the reconstruction of three-dimensional surface, territory from two-dimentional sketch reconstruction of three-dimensional surface, described two-dimentional sketch comprises at least one pen, described at least one pen is assigned vector field, said method comprising the steps of: insert described vector field to form fine and close vector field; Use described fine and close vector field to rebuild described surface; Wherein, inserting described vector comprises: the Nonlinear Functional that farthest reduces described vector field, described vector field has irrotationality constraint (curl-free constraint) on described fine and close vector field, the Nonlinear Functional of described vector field comprises a plurality of and the item of the upper definition of described at least one pen defined on described territory, wherein, farthest reduce the Nonlinear Functional of described vector field by the augmented Lagrangian method.

A first aspect of the present invention has superiority, because it is a kind of fast and effectively mode on reconstruction of three-dimensional surface.And it has realized the reconstructed surface of the described allocation vector of accurate reflection field.In the situation that do not produce surface deformation, it also helps the surface details of preserving described reconstructed surface.Therefore, can realize the structures that need more.And, use the augmented Lagrangian method to solve described Nonlinear Functional, the augmented Lagrangian method can solve described functional quickly.

The present invention alternately is expressed as the computer system (for example, the surface model building system based on sketch) for carrying out this method.Described computer system can be integrated in one with the device for recording sketch, and wherein, described device can further comprise user interface, and described user interface is for generating or extract sketch.The present invention can also be expressed as computer program, such as, being recorded in the computer program on tangible computer media, described product comprises can be by the programmed instruction of computer system operation in order to carry out each step of described method.

The accompanying drawing explanation

Below with reference to following accompanying drawing, only with the form of example, embodiments of the invention are illustrated.

Fig. 1 illustrates the process flow diagram for the method from 2D sketch reconstruction 3D surface according to the embodiment of the present invention;

Fig. 2 (a) illustrates user's sketch, and Fig. 2 (b)-(d) illustrates the effect on user's sketch shown in Fig. 2 (a) by each step application in method shown in Fig. 1;

Fig. 3 (a) illustrates user's sketch, and Fig. 3 (b)-(d) illustrates the effect on user's sketch shown in Fig. 3 (a) by each step application in method shown in Fig. 1;

Fig. 4 (a)-(c) illustrates the example of the different stroke types of using in method shown in Fig. 1;

Fig. 5 (a) illustrates the HSV coloured picture, and Fig. 5 (b)-(d) illustrates the vector distributed along the exemplary stroke in method shown in Fig. 1;

Fig. 6 (a)-(c) illustrates the vector that is assigned to user's sketch, and Fig. 6 (d)-(f) illustrates respectively and uses the vector distributed in Fig. 6 (a)-(c) by the surface of the reconstruction of method shown in Fig. 1;

Fig. 7 (a)-(b) illustrates respectively the concept map of carinate stroke and paddy shape stroke;

The reconstructed surface that Fig. 8 (a)-(c) illustrates basic strokes, stroke interested and is respectively used to great-jump-forward stroke, raised water stroke and tilting stroke;

Fig. 9 (a)-(d) illustrates the example how weighting function of using in method shown in Fig. 1 affects reconstructed surface;

Figure 10 (a)-(f) illustrates another the example how weighting function of using in method shown in Fig. 1 affects reconstructed surface;

Figure 11 (a)-(f) illustrates on the vector inserted in method shown in Fig. 1 and forces the example how the irrotationality state affects reconstructed surface;

Figure 12 (a)-(c) illustrates respectively the relative residual error that in the process of iteration of using in contrast Fig. 1 method, iterations is drawn, the relative error of Lagrange multiplier, and the numerical value energy value;

Figure 13 (a) illustrates the indexed rule of variable in the augmentation Lagranian functional for method shown in Fig. 1 is used, and Figure 13 (b) illustrates execution graph 13(a in the discrete calculation territory) in the example of rule;

Figure 14 (a) and the surface that 14(b) illustrates respectively user's sketch and rebuild from user's sketch by method shown in Fig. 1, Figure 14 (c) and 14(d) illustrate respectively and there is the user's sketch that adds stroke and the reconstructed surface that has carried out modification corresponding to this interpolations stroke;

Figure 15 (a) and (e) illustrate respectively user's sketch and distribute to the initialization vector of user's sketch, Figure 15 (b) and (f), (c) and (g) and (d) and (h) illustrate respectively method shown in fine and close vector field and use LUMO method, ShapePalettes method and Fig. 1 from the surface of user's sketch reconstruction;

Figure 16 (a) illustrates user's sketch, and Figure 16 (b)-(d) illustrates respectively and uses the surface that shown in LUMO method, ShapePalettes method and Fig. 1, method is rebuild from user's sketch;

The vector that Figure 17 (a)-(c) illustrates respectively user's sketch, distribute to user's sketch is used the surface that shown in Fig. 1, method is rebuild from user's sketch;

Figure 18 (a)-(d) illustrates user's sketch and each uses the surface that shown in Fig. 1, method is rebuild from user's sketch;

Figure 19 (a)-(b) illustrates another the example how weighting function of using in method shown in Fig. 1 affects reconstructed surface, and Figure 19 (c)-(d) illustrates the example how different stroke types affect the reconstructed surface in method shown in Fig. 1;

Figure 20 (a)-(d) illustrates the framework that uses the exemplary animation sequence that shown in Fig. 1, method creates.

Embodiment

With reference to Fig. 1, it illustrates each step of the method 100 of the embodiment of the present invention, and method 100 is rebuild the 3D surface from comprising the 2D sketch of at least one.

In step 102, draw stroke to form user's sketch, then the type based on stroke is classified to these strokes.In step 104, along the stroke allocation vector in user's sketch.Next, in step 106, insert the vector of distribution to form fine and close vector field, in step 108, use fine and close vector field to rebuild the 3D surface.

User's sketch can comprise by paint pen in step 102 draws the basic 2D shape formed.In one example, adopt the simple line chart of the feature that has reflected required 3D surface to form sketch.In another example, ignore shade (shading) in user's sketch, highlighted, hachure stroke and strokes and dots.The stroke of user's sketch can be used vector format to be recorded in method 100 as input.This record can utilize the graphics module of the animation system CACAni based on vector to carry out.After recording each stroke, the user can be by other strokes of painting, follow the trail of the stroke record or the reference mark adjustment of recorded stroke is revised to its sketch.

Fig. 2 (a) illustrates the example user sketch with the form of the line chart of ancient china phoenix, and wherein, stroke is and form that present clean vector stroke drawing with the style of profile diagram.Fig. 2 (b)-(d) illustrates respectively step 104,106 and 108 is applied to the effect on user's sketch shown in Fig. 2 (a).Especially, Fig. 2 (b) illustrates the pointer figure (needle map) of part sketch shown in Fig. 2 (a), wherein, pointer figure has shown the initialization vector on the stroke be automatically assigned in this part sketch, Fig. 2 (c) illustrates the vector will distributed and inserts the fine and close vector field obtained after Fig. 2 (b), and Fig. 2 (d) illustrates and uses the surface that shown in Fig. 2 (c), fine and close vector field is rebuild.

Fig. 3 (a) illustrates another example user sketch.Similarly, Fig. 3 (b)-(d) illustrates respectively step 104,106 and 108 is applied to the effect after user's sketch shown in Fig. 3 (a), wherein, Fig. 3 (b) illustrates the vector that is assigned to the stroke in sketch, Fig. 3 (c) illustrates the fine and close vector field of the vector acquisition distributed from Fig. 3 (b), and Fig. 3 (d) illustrates the surface of using the fine and close vector field in Fig. 3 (c) to rebuild.

Below will illustrate in further detail step 102-108.

step 102: draw stroke to form user's sketch and these strokes are classified

In step 102, draw stroke to form user's sketch.Can be that the user provides dissimilar stroke in order to form user's sketch.

The stroke type that the user draws in step 102 has been determined and has been treated the surface geometry (geometry) again created from these strokes.In step 102, then according to the type of stroke, the stroke in user's sketch is classified.In one example, the user can imagine intuitively and treat the surface geometry again created from these stroke types from the stroke type that offers them.The nearest investigation that artists express 3D shape [33] by setting-out shows, people can detect jump, carinate, paddy shape, protuberance and inclination easily from the 3D surface.Therefore, in one example, the stroke type that offers the user comprises great-jump-forward stroke, carinate stroke, paddy shape stroke, raised water stroke and tilting stroke, and these strokes are for rebuilding these different partial structurtes.Stroke type can further comprise regular stroke, and regular stroke is for determining the global shape of reconstructed surface.In other words, each stroke in user's sketch can classify as following one of them: regular stroke, great-jump-forward stroke, carinate stroke, paddy shape stroke, raised water stroke or tilting stroke.In one example, stroke (for example, draw as shown in Figure 2 (a) shows), and default setting is regular stroke with the style of profile diagram.In this example, the user can edit sketch by the stroke type of selecting one or more regular strokes and changing selected stroke.This is conducive to fast direct ground connection and forms and edit sketch.

The different stroke types that are used to form user's sketch (regular stroke 402(solid black lines) that provide in step 102, carinate stroke 404(Grey Point solid line are provided Fig. 4 (a)-(c)), paddy shape stroke 406(black color dots solid line), great-jump-forward stroke 408(black dotted lines), raised water stroke 410(Grey Point solid line) and tilting stroke 412(black color dots solid line)) example, and each surface geometry of rebuilding from these stroke types.As shown in Fig. 4 (a)-(c), comprise discontinuous point from the geometric figure of raised water and the formation of tilting stroke at surface normal, regular stroke changes smooth change in height map.Note, as if, although regular stroke has produced different structures in Fig. 4 (a)-(c), this is to be caused by the smudgy of visual angle, and in fact, the surface geometry that regular stroke creates is the same.

step 104: along the stroke allocation vector in user's sketch

In step 104, each stroke in user's sketch is distributed a plurality of vectors.The vector distributed (except the vector distributed along the great-jump-forward stroke) means that the 3D surface normal is in the projection had on the 2D plane of sketch.

In one example, at first along each stroke in user's sketch, automatically distribute a plurality of initialization vectors.The term of using herein " automatically " shows, although artificial participation can make this process start, does not just need artificial participation when carrying out this process.Useful predetermined scheme is carried out the distribution of initialization vector.In this example, the tangent line of initialization vector and stroke (that is, the stroke tangent line) quadrature, the amplitude of each initialization vector can identical or different (for example,, in the situation that the T stroke, this be described in detail with reference to Fig. 5 (d)).Then can revise initialization vector (by revising its amplitude and/or direction) to mean more exactly required surface geometry.The user for example can use based on it keyboard or mouse manually to carry out modification to the imagination on required surface.This means that than input the numerical value of required surface elevation is more directly perceived.And the user can, before determining and how revising vector, rebuild the 3D surface from initialization vector.Replacedly, also can user's setting or predefined scheme based on prestrain automatically carry out modification.Note, the modification of initialization vector is random.

Fig. 5 (a) illustrates the HSV(hue saturation value) output of the black and white of coloured picture, wherein, V=1.This figure can be used for the vector that diagram is distributed along the stroke in user's sketch as shown in Fig. 5 (b)-(d).Fig. 5 (b) illustrates the vector of original allocation, and these vectors are vertical with the stroke tangent line and have identical amplitude, and Fig. 5 (c) illustrates the setting based on specific user definition and adjusted the initialization vector of amplitude and direction.Fig. 5 (d) illustrates the stroke of how processing in the combination of T junction.For the T junction, initialization vector can be assigned as to make and approaches the T junction and reduce along the amplitude of vector of the visible part of sealing stroke 502, to avoid impact to approach the T junction and along the vector of another stroke 504.In order to distribute the initialization vector with different amplitudes, can automatically detect this T junction.

The vector that is assigned to each stroke in step 102 has geometric meaning.For example, (the great-jump-forward stroke is illustrated in reconstructed surface will form the step structure formula, surface elevation is interrupted or reduces suddenly) jump, the direction of the vector distributed along this great-jump-forward stroke and direction (that is, surface elevation should reduce in which side of stroke) and the amplitude that amplitude means respectively jump to be formed.And, the stroke that belongs to remaining stroke type is illustrated in reconstructed surface and will forms oblique line (, surface elevation constantly reduces), the direction of the vector distributed along each these strokes and direction (that is, surface elevation will reduce in which side of stroke) and the degree of tilt that amplitude means respectively oblique line to be formed.Larger amplitude means oblique line that need to be steeper, in other words, needs larger surface elevation reduction rate.

The rule stroke is to determine the stroke of the global shape of reconstructed surface.In one example, based on the acquiescence height map, initialization vector is assigned on regular stroke.Fig. 6 (a) illustrates the sketch with the initialization vector distributed along regular stroke, and Fig. 6 (d) illustrates the 3D surface of the vector reconstruction of using in Fig. 6 (a).Fig. 6 (b) only illustrates the vector distributed along the stroke that means the wing of nose 602 in sketch.With corresponding vector in Fig. 6 (a), compare, the vector magnitude in Fig. 6 (b) is larger.Fig. 6 (e) illustrates the 3D surface of the vector reconstruction of using in Fig. 6 (b), wherein, can find out, larger vector magnitude produces the higher wing of nose.Fig. 6 (c) only illustrates the vector be assigned on the stroke that means left ear 604.With corresponding vector in Fig. 6 (a), compare, these vectors are rotated counterclockwise 15 degree.Fig. 6 (f) illustrates the 3D surface of the vector reconstruction of using in Fig. 6 (c), wherein, can find out, the rotation of vector produces the surface of distorted shape, thereby produces the face of distortion.

Carinate (or paddy shape) stroke means on reconstructed surface to need significantly carinate (or paddy shape).In one example, carinate (or paddy shape) stroke is the regular stroke form of two adjacent, parallel, as shown in Fig. 7 (a)-(b).Note, Fig. 7 (a)-(b) is concept map, and the space between two parallel regular strokes is normally sightless.In fact, carinate (or paddy shape) stroke is not to adopt this form usually in user's sketch.Draw the stroke that is used to form carinate (or the paddy shape) stroke in user's sketch and preferably be the single line strips, for example, because this is more prone to (use for the user, as shown in Figure 4, carinate (or paddy shape) stroke gray point solid line (or black color dots solid line)).As shown in Fig. 7 (a)-(b), distribute to initialization vector and the stroke tangent line quadrature of two parallel regular strokes of carinate (or paddy shape) stroke, and described initialization vector or be set to form in the direction of mutually repelling carinate, or be assigned as in the direction attracted each other and form the paddy shape.In other words, the direction of the initialization vector distributed along carinate stroke is different from the direction of the initialization vector distributed along paddy shape stroke, differs 180 degree.In one example, the user only can change the amplitude of the initialization vector of distributing to carinate (or paddy shape) stroke, and can not change its direction.If the user has increased the amplitude of vector, will in reconstructed surface, create out so steeper carinate (or darker paddy shape).

Great-jump-forward, raised water and tilting stroke are of great use in reconstructed surface, generating exquisite structure.These structures can form at the top on the given surface as shown in Fig. 8 (a)-(c).The surface of appointment can be the initial surface that the regular stroke from user's sketch, carinate stroke and/or paddy shape stroke are rebuild.

The reconstructed surface that Fig. 8 (a)-(c) illustrates basic strokes, stroke interested (SOI) and is respectively used to great-jump-forward stroke, raised water stroke and tilting stroke.Each great-jump-forward stroke, raised water stroke and tilting stroke comprise that basic strokes and an adjacent S OI(are in the situation that raised water stroke or tilting stroke) otherwise two adjacent S OI(in the situation that the great-jump-forward stroke).In Fig. 8 (a)-(c) first, second, and third row is concept map.Especially, first row illustrates does not have jump, the given surface of protuberance and incline structure, second row illustrates the variation of surface normal, and surface normal changes to form corresponding structure along basic strokes, and the 3rd row illustrates the surface that has jump, swells and tilt.The 4th row of Fig. 8 (a)-(c) illustrate there is jump, the reconstruction 3D surface of protuberance and incline structure.Point solid line, solid line and dotted arrow in Fig. 8 (a)-(c) means respectively the normal direction of the regular stroke in user's sketch, the given surface rebuild from regular stroke and the basic strokes of great-jump-forward, raised water and tilting stroke.

As shown in Fig. 8 (a), the great-jump-forward stroke comprises two SOIs parallel and adjacent with its basic strokes.In order to form required jump, when starting vector assignment is arrived to basic strokes, reduce surface elevation, in order to form the oblique line of correct angle.Note, in this example, the amplitude that is assigned to the vector on basic strokes does not mean the degree of tilt of oblique line.On the contrary, it means the size (that is, the decrease of surface elevation) of required jump in reconstructed surface.If there is a given surface that has jump, so, it is upper that the vector identical with the surface normal on given surface just is assigned to SOI, means the geometric figure on given surface.If there is no this given surface, so, the vector distributed along SOI is initialized as zero.

As shown in Fig. 8 (b)-(c), raised water (or tilting) stroke comprises a SOI parallel and adjacent with its basic strokes.Raised water (or tilting) stroke means to form oblique line in reconstructed surface, and the amplitude of distributing to the vector of basic strokes means required oblique line slope.Similarly, if there is a given surface that has protuberance (or inclination), so, the vector assignment identical with the surface normal on given surface is to SOI.If there is no this given surface, so, the vector distributed along SOI is initialized as zero.

Below to use different strokes to rebuild the instantiation procedure on 3D surface.At first, user's service regeulations stroke, carinate stroke and/or paddy shape stroke draw the general shape on required surface.Then initialization vector is assigned on these strokes, and from the vector reconstructed surface of these distribution.Then, the user revises the amplitude of initialization vector and/or the reconstructed surface that direction needs more with acquisition based on these reconstructed surfaces.Then draw great-jump-forward, raised water and tilting stroke to form detailed geometric figure structure on reconstructed surface.Then, by vector assignment to basic strokes and as Fig. 8 (a)-(c) as shown in the SOI of these great-jump-forwards, raised water and tilting stroke upper, wherein, be assigned to the surface normal of the vector representation of SOI from surperficial (that is, the given surface) projection of original reconstruction.

In one example, the user only can rotate the initialization vector distributed along regular stroke.This is because, although can not know intuitively the effect of this rotation by rotating the geometric figure that adjust reconstructed surface along the initialization vector of other stroke type distribution, especially for those, do not understand the user of the mathematical meaning of vector.Therefore, in order to allow, method 100 is more directly perceived to be used with being more prone to, the initialization vector that does not preferably allow rotation to distribute along other stroke type.

step 106: insert the vector field of distribution to form fine and close vector field

In step 106, be inserted in the vector of distribution in step 104 to obtain fine and close vector field.

Sparse stroke in user's sketch can be expressed as wherein,

it is the computational fields of rebuilding the 3D surface.The vector distributed along stroke Γ can vector field

form express.In step 106, insert vector field n ^*to obtain fine and close vector field n=(n on computational fields Ω ₁, n ₂) ^t.3D surface and height map z=I (x to be reconstructed ₁, x ₂) be associated, wherein, z is the point (x had in the 2D plane of sketch ₁, x ₂) desired height located.The vector n distributed along stroke (except the great-jump-forward stroke) ^*can regard the projection of surface normal vector on the 2D plane of 3D surface (thering is desired height figure z) as.Because the direction of the surface normal vector on 3D surface can be expressed as

so vector n distributed along stroke (except the great-jump-forward stroke) ^*can be expressed as-▽ I.

In vector calculates, if just relate to simply computational fields

so, and if only if

so that during ▽ I=n, equation (1) is just set up.Equation (1) means the irrotationality condition, and irrotationality is adjusted in the visible surface reconstruction and is commonly referred to integrability condition.

$\&dtri;\&times;n\&equiv;{\&PartialD;}_2{n}_1-{\&PartialD;}_1{\mathrm{<figure-callout\; id="2"\; label="n"\; filenames="HDA00003765143700111.png,HDA00003765143700121.png"\; state="\{\{state\}\}">n</figure-callout>}}_2=0---\left(1\right)$

Because will rebuild and height map z=I (x ₁, x ₂) surface that is associated, the degree of tilt of height map is applicable to the vector n inserted, so preferably meet the irrotationality condition on (or fine and close) the vector field n inserted.

Use total variation (TV) rule method and the H of weighting ¹the combination of rule method performs step 106, and the total variation rule method is for preserving the acutance of rebuilding the 3D surface, H ¹rule method is for preserving the smoothness of rebuilding the 3D surface.Especially, this energy minimization functional that can have a constraint of irrotationality as shown in equation (2) by use is carried out.Equation (2) illustrates and at utmost reduces vector field n ^*nonlinear Functional, Nonlinear Functional comprises a plurality of and the upper item defined of stroke Γ of the upper definition of territory Ω.In other words, equation (2) is implemented non-linear vector insertion.Note, equation (2) will be separated by the constraint of the irrotationality on fine and close vector field n.

$\underset{n}{\min }\{{\mathrm{\&epsiv;}}_v\left(n\right)\&equiv;{\&Integral;}_{\mathrm{\&Omega;}}(1-g){|\&dtri;n|}_F+{\&Integral;}_{\mathrm{\&Omega;}}g{|\&dtri;n|}_{\mathrm{<figure-callout\; id="2"\; label="F"\; filenames="HDA00003765143700111.png,HDA00003765143700121.png"\; state="\{\{state\}\}">F</figure-callout>}}^2+\mathrm{\&eta;}{\&Integral;}_{\mathrm{\&Gamma;}}|n-n^{*}|\},$ Wherein, ▽ * n=0 (2)

It is TV rule method and H that ability in equation (2) minimizes functional ¹the weighted combination of rule method.In equation (2), fine and close irrotational vector field n is by farthest reducing energy function ε _vand the vector field n from distributing along stroke Γ (n) ^*obtain.| .| is R ²in euclideam norm, | .| _fit is Fu Luobin Nice (Frobenius) norm as shown in following equation (3).Note, integrated territory Γ comprises sealing and open stroke.In other words, integrated territory Γ comprises all SOI and the basic strokes in great-jump-forward, raised water and tilting stroke.

${|\&dtri;n|}_{\mathrm{<figure-callout\; id="2"\; label="F"\; filenames="HDA00003765143700111.png,HDA00003765143700121.png"\; state="\{\{state\}\}">F</figure-callout>}}^2={\left|\left(\begin{array}{c}{\&dtri;\mathrm{<figure-callout\; id="1"\; label="n"\; filenames="HDA00003765143700111.png,HDA00003765143700121.png"\; state="\{\{state\}\}">n</figure-callout>}}_1\\ {\&dtri;\mathrm{<figure-callout\; id="2"\; label="n"\; filenames="HDA00003765143700111.png,HDA00003765143700121.png"\; state="\{\{state\}\}">n</figure-callout>}}_2\end{array}\right)\right|}_{\mathrm{<figure-callout\; id="2"\; label="F"\; filenames="HDA00003765143700111.png,HDA00003765143700121.png"\; state="\{\{state\}\}">F</figure-callout>}}^2\&equiv;\underset{j=1}{\overset{2}{\mathrm{\&Sigma;}}}\underset{i=1}{\overset{2}{\mathrm{\&Sigma;}}}{\left({\&PartialD;}_j{n}_i\right)}^2---\left(3\right)$

In equation (2), first ∫ _Ω(1-g) | ▽ n| _fmean the TV rule method, and second

mean H ¹rule method.The 3rd η ∫ _Γ| n-n ^*| (wherein, η is normal number) is the n on the Γ preserved in the vector field n inserted ^*the punishment parameter.

Usually, H ¹rule method is rebuild more level and smooth surface, and the TV rule method is rebuild more sharp-pointed surface.This is because the discontinuous point in the vector field of distributing along stroke usually can insert well in the Ω of territory under the TV rule method, and at H ¹under rule method, discontinuous point is fuzzy entering in the Ω of territory.Therefore, mean first of TV rule method for generation of nonlinear diffusion in equation (2) and preserve discontinuous point, for example, insert in the folding line in vector field or jump in discontinuous point, and in equation (2) expression H ¹second discontinuous point that for generation of linearity, spreads and make to insert in vector field of rule method is fuzzy.

It is advantageous preserving discontinuous point with the TV rule method.Because, other algorithm (such as, the algorithm in [20]) need to mean the information of discontinuous point in ΩZhong position, territory, the TV rule method does not need this information.Because usually be difficult to only just find from the vector distributed along stroke the position of discontinuous point, so use the TV rule method in equation (2) that the enforcement of method 100 is more prone to.

In equation (2), use can come Weighted T V rule method and H with weighting function (or indicator function) g of arbitrary value in 0 and 1 ¹rule method.The value of g (that is, weight) is determined smoothness and the acutance of reconstructed surface,, determines the texture of rebuilding the 3D surface that is.Therefore, by by TV rule method and H as shown in equation (2) ¹rule method and different g value combination can generate various surface geometries in reconstructed surface.Note, when g=1, only implement H ¹rule method, and, when g=0, only implement the TV rule method.In one example, the required texture of the g value of sketch based on surface to be reconstructed set, and by using this g value Weighted T V and H ¹the vector of distribution is inserted in the combination of rule method.

The g value of sketch (comprising the space between each stroke in each stroke and sketch) can be by the user by being used, and for example drawing instrument, set.In one example, drawing instrument is for to sketch, coating the color that a kind of pre-variable Alpha (alpha) value is associated, and the g value of sketch can be set by changing alpha value.In order to carry out this operation, g value ([0,1]) can be composed alpha value (from [0,1] to 0,255] or from [0,1] to [255,0]).

In one example, after having set the g value of sketch, each stroke be categorized as in sketch based on stroke is revised the g value.Consider TV rule method as above and H ¹the difference of rule method, for great-jump-forward, carinate, paddy shape, raised water and tilting stroke, can be revised as 0(by the g value in order to preserve the discontinuous point of resulting structures), for regular stroke, g value can be revised as to 1(so that the generation smooth surface).

Replacedly, user's sketch can be divided into a plurality of parts, and the required texture on surface that can be based on treating to rebuild from this part and be that various piece is set different weights.In one example, in first, g is made as 0 in order to preserve the discontinuity of the surface slope in first, and in second portion, g is made as 1 in order to generate smooth surface in second portion.Various piece can be coated a kind of color be associated with alpha value, can change alpha value to set the weight of this part.Similarly, the g value can be composed alpha value (from [0,1] to 0,255] or from [0,1] to [255,0]), and can revise according to the classification of stroke the g value of each stroke in various piece.

Fig. 9 (a)-(d) and 10(a)-how the g value that (d) illustrates weighting function to affect and rebuild the 3D surface.

Fig. 9 (a) and the surface that 9(b) illustrates respectively as g=1 and rebuild from user's sketch during as g=0.Fig. 9 (c) illustrates user's sketch is divided into to two parts: left-half and right penumbra part.G=1 is set and at right penumbra, the reconstructed surface of g=0 in generating Fig. 9 (d) partly is set in the left-half of sketch.As shown in Fig. 9 (d), reconstructed surface be Fig. 9 (a) and 9(b) in the natural combination of various piece of reconstructed surface.

Figure 10 (a) and 10(b) illustrate respectively the user's sketch that comprises regular stroke and distribute to the initialization vector of stroke.Note, distribute to the vector in the cross section in rectangle parallel with the stroke tangent line in cross section.In the initialization vector field, there are four discontinuous points at place, four of rectangle angles outside, and three discontinuous points are arranged on cross section.These seven discontinuous points mean by the arrow in Figure 10 (b).Figure 10 (d) illustrates by the g by whole sketch and is made as 0(, only by using the TV rule method) surface that rebuilds.As shown in Figure 10 (d), the discontinuous point in the initialization vector field well inserts and is sharp-pointed carinate shape.Figure 10 (e) illustrates by the g by whole sketch and is made as 0(, by only using H ¹rule method) surface rebuild.As shown in Figure 10 (e), corresponding to the vector inserted, obtain smooth surface in zone.Figure 10 (c) illustrates user's sketch is divided into to two parts: veined part and do not have a veined part.Veined part is made as to g=0, will have veined part to be made as g=1, this will generate the reconstructed surface in Figure 10 (f).As shown in Figure 10 (f), reconstructed surface be Figure 10 (d) and 10(e) in the natural combination of various piece of reconstructed surface.

Modeling in equation (2) retrains to carry out by the irrotationality of forcing on fine and close vector field n.This is conducive to rebuild height map I, wherein ▽ I=n.It is conducive to insert sparse vector equally on large territory.Figure 11 (a)-11(f) illustrate and there is the irrotationality constraint of forcing on the vector inserted.Figure 11 (a) illustrates the user's sketch that comprises regular stroke, and Figure 11 (b) illustrates the initialization vector distributed along the stroke of user's sketch.Figure 11 (c) and Figure 11 (d) illustrate the reconstructed surface that there is no irrotationality constraint of acquisition, and (wherein, Figure 11 (c) illustrates and uses H ¹rule method obtains reconstructed surface, and Figure 11 (d) illustrates the reconstructed surface that uses the TV rule method to obtain).Figure 11 (e) and Figure 11 (f) illustrate the reconstructed surface with irrotationality constraint of acquisition, and (wherein, Figure 11 (e) illustrates and uses H ¹the reconstructed surface that rule method obtains, Figure 11 (f) illustrates the reconstructed surface that uses the TV rule method to obtain).Respectively by Figure 11 (c) and 11(d) and Figure 11 (e) and 11(f) compare, can find out, if having the irrotationality constraint, weighting function just has larger impact to reconstructed surface.And, if having the irrotationality constraint, reconstructed surface can reflect the initialization vector settings more exactly, and can obtain the details of sharpened surface.

Yet it is very difficult will applying irrotationality constraint and make it meet whole territory Ω, because the discontinuous point place that irrotationality constrains in the vector field of insertion can be interfered usually.Below described and overcome this difficult illustrative methods.

farthest reduce the Augmented Lagrange method of energy function in equation (2)

In one example, farthest reduce energy function (that is, the Nonlinear Functional of vector field) with Augmented Lagrange method in equation (2).This has overcome the approximately intrafascicular difficulty of application irrotationality, and makes the irrotationality constraint almost meet all places in the Ω of territory.Below in detail the method will be described in detail.

Use and have the operator splitting method of two variablees as shown in equation (4), equation (2) just changes constrained minimization problem into, as shown in equation (5).

$P=\left(\begin{array}{c}{p}_1\\ {\mathrm{<figure-callout\; id="2"\; label="p"\; filenames="HDA00003765143700111.png,HDA00003765143700121.png"\; state="\{\{state\}\}">p</figure-callout>}}_2\end{array}\right)=\left(\begin{array}{cc}{p}_{11}& p_{12}\\ p_{21}& p_{22}\end{array}\right)$ With $s=\left(\begin{array}{c}{s}_1\\ s_2\end{array}\right)---\left(4\right)$

$\underset{n}{\min }{\&Integral;}_{\mathrm{\&Omega;}}(1-g){\left|P\right|}_F+{\&Integral;}_{\mathrm{\&Omega;}}g{\left|P\right|}_{\mathrm{<figure-callout\; id="2"\; label="F"\; filenames="HDA00003765143700111.png,HDA00003765143700121.png"\; state="\{\{state\}\}">F</figure-callout>}}^2+\mathrm{\&eta;}{\&Integral;}_{\mathrm{\&Gamma;}}|s-n^{*}|$

Wherein, P=▽ n, s=n, and ▽ * n=0 (5)

Finally change for the constraint in solve an equation (5), formed the augmentation Lagranian functional L as shown in equation (6) _v(s, n, P; λ _f, λ _c, Λ _r), wherein, c _f, c _cand c _rfor just punishing parameter, λ _f, λ _cand Λ _rfor Lagrange multiplier.If the weighting function g in equation (2) is set to 1 on Ω, so just be necessary to use c _r, Λ _rand P.

$L_v(.)\&equiv;{\&Integral;}_{\mathrm{\&Omega;}}(1-g){\left|P\right|}_F+g{\left|P\right|}_{\mathrm{<figure-callout\; id="2"\; label="F"\; filenames="HDA00003765143700111.png,HDA00003765143700121.png"\; state="\{\{state\}\}">F</figure-callout>}}^2+\mathrm{\&eta;}{\&Integral;}_{\mathrm{\&Gamma;}}|s-n^{*}|$

$+{\&Integral;}_{\mathrm{\&Omega;}}{\mathrm{\&Lambda;}}_r\&CenterDot;(P-\&dtri;n)+\frac{{\mathrm{<figure-callout\; id="2"\; label="c\; r"\; filenames="HDA00003765143700111.png,HDA00003765143700121.png"\; state="\{\{state\}\}">c</figure-callout>}}_r}{2}{\&Integral;}_{\mathrm{\&Omega;}}{|P-\&dtri;n|}_{\mathrm{<figure-callout\; id="2"\; label="F"\; filenames="HDA00003765143700111.png,HDA00003765143700121.png"\; state="\{\{state\}\}">F</figure-callout>}}^2+{\&Integral;}_{\mathrm{\&Omega;}}{\mathrm{\&lambda;}}_f\&CenterDot;(s-n)+\frac{{\mathrm{<figure-callout\; id="2"\; label="c\; f"\; filenames="HDA00003765143700111.png,HDA00003765143700121.png"\; state="\{\{state\}\}">c</figure-callout>}}_f}{2}{\&Integral;}_{\mathrm{\&Omega;}}{|s-n|}^2$

$+{\&Integral;}_{\mathrm{\&Omega;}}{\mathrm{\&omega;}}_c{\mathrm{\&lambda;}}_c(\&dtri;\&times;n)+\frac{{\mathrm{<figure-callout\; id="2"\; label="c\; c"\; filenames="HDA00003765143700111.png,HDA00003765143700121.png"\; state="\{\{state\}\}">c</figure-callout>}}_c}{2}{\&Integral;}_{\mathrm{\&Omega;}}{(\&dtri;\&times;n)}^2---\left(6\right)$

ω _cequation (6) is a weighting function.Carry out adaptive selection ω according to geometry to be reconstructed _c, and the geometry of rebuilding correspondingly depends on stroke type.In one example, because deviate from the irrotationality constraint on the discontinuous point of vector field n in surface elevation and degree of tilt inserted, therefore, if carinate, paddy shape, raised water, tilting and great-jump-forward stroke, ω _cbe made as 0, if not ω _cjust be made as 1.In one example, weighting function ω _caccording to the priori of the stroke type on user's sketch, automatically determine.Note, although work as ω _c=1 o'clock, the folding line structure that irrotationality retrains may appear deviating from, still, because in equation (2), existing of TV rule method (seen Fig. 9 (d), so reduced the possibility of surface deformation.In one example, because ω _csmooth function, so by ω _cbe set to have the convolution of the 2D gaussian kernel of smoothing factor 2.In other words, the Gauss's linear filtering that has a scale parameter 2 by execution obtains ω _c.Also available identical core is as the weighting function g in equation (2).There is the ω that there is no variable in equation (6) _cbe useful, be necessary for the stroke that deviates from the irrotationality constraint and select quite little penalty parameter c _c.And little c _ccan make after a while the convergence of the iterative process that will describe slow, be therefore unpractical.

Be below an exemplary algorithm, this exemplary algorithm comprises iterative process, so that by farthest reducing Lagrange multiplier and farthest reducing variable s, n and P finds out the saddle point of the augmentation Lagranian functional in equation (6).

algorithm 1: the Augmented Lagrange method of equation (6)

Step 1: by s ⁰, n ⁰, P ⁰,

with

be initialized as 0.Variable k(means to exist the iteration of algorithm) also be initialized as 0.

Step 2: the k value is increased to 1, if the augmentation Lagranian functional is calculated with regard to the Method for minimization with replacing in k>=1

approximate minimizer (s ^k, n ^k, P ^k).This step is carried out according to equation (7), wherein, s, n, P treats minimized value,

$(s^k,n^k,p^k)\&ap;\arg \min L_v(s,n,p;{\mathrm{\&lambda;}}_f^{k-1},{\mathrm{\&lambda;}}_c^{k-1},{\mathrm{\&Lambda;}}_r^{k-1})---\left(7\right)$

Step 3: use following equation (8)-(10) to upgrade Lagrange multiplier.

${\mathrm{\&lambda;}}_f^k={\mathrm{\&lambda;}}_f^{k-1}+c_f(s^k-n^k)---\left(8\right)$

${\mathrm{\&lambda;}}_c^k={\mathrm{\&lambda;}}_c^{k-1}+\frac{c_c}{{\mathrm{\&omega;}}_c}(\&dtri;+n^k)---\left(9\right)$

${\mathrm{\&Lambda;}}_r^k={\mathrm{\&Lambda;}}_r^{k-1}+c_r(P^k-{\&dtri;n}^k)---\left(10\right)$

Note, if at a some place, ω _c=0, so at this some place

Step 4: use equation (11) to calculate relative residual error

if relative residual error

ε is large for the ratio error scope, with regard to repeating step 2-4.Note, residual error does not rely on size or the punishment parameter in territory relatively, and therefore, it is suitable for determining whether to stop iterative step.

$R_{\mathrm{vi}}^k\&equiv;\frac{1}{\left|\mathrm{\&Omega;}\right|}{\left|\right|{\tilde{R}}_{\mathrm{vi}}^k\left|\right|}_{{\mathrm{<figure-callout\; id="1"\; label="L"\; filenames="HDA00003765143700111.png,HDA00003765143700121.png"\; state="\{\{state\}\}">L</figure-callout>}}^1},\&ForAll;i\&Element;\left\{\mathrm{1,2,3}\right\}---\left(11\right)$

Wherein, the L on Ω ¹norm, | Ω | be the zone in territory.

$({\tilde{R}}_{v1}^k,{\tilde{R}}_{v2}^k,{\tilde{R}}_{v3}^k)=(s^k-n^k,\&dtri;\&times;n^k,P^k-{\&dtri;n}^k)---\left(12\right)$

In one example, the relative error of Lagrange multiplier is also calculated in the step 4 of algorithm 1, to determine whether to stop iteration (that is, whether should repeating step 2-4).These relative errors are as shown in equation (13) and calculate.In another example, in order to determine whether to stop iterative step, also evaluation energy.The numerical value energy is as shown in equation (14) and calculate.Similarly, relative error and/or numerical value energy value and corresponding error range are contrasted, in order to determine whether to stop iteration.

$L_{v1}^k=\frac{{\left|\right|{\mathrm{\&lambda;}}_f^k-{\mathrm{\&lambda;}}_f^{k-1}\left|\right|}_{{\mathrm{<figure-callout\; id="1"\; label="L"\; filenames="HDA00003765143700111.png,HDA00003765143700121.png"\; state="\{\{state\}\}">L</figure-callout>}}^1}}{{\left|\right|{\mathrm{\&lambda;}}_f^{k-1}\left|\right|}_{{\mathrm{<figure-callout\; id="1"\; label="L"\; filenames="HDA00003765143700111.png,HDA00003765143700121.png"\; state="\{\{state\}\}">L</figure-callout>}}^1}},$ $L_{v2}^k=\frac{{\left|\right|{\mathrm{\&lambda;}}_c^k-{\mathrm{\&lambda;}}_c^{k-1}\left|\right|}_{{\mathrm{<figure-callout\; id="1"\; label="L"\; filenames="HDA00003765143700111.png,HDA00003765143700121.png"\; state="\{\{state\}\}">L</figure-callout>}}^1}}{{\left|\right|{\mathrm{\&lambda;}}_c^{k-1}\left|\right|}_{{\mathrm{<figure-callout\; id="1"\; label="L"\; filenames="HDA00003765143700111.png,HDA00003765143700121.png"\; state="\{\{state\}\}">L</figure-callout>}}^1}},$ $L_{v3}^k=\frac{{\left|\right|{\mathrm{\&Lambda;}}_r^k-{\mathrm{\&Lambda;}}_r^{k-1}\left|\right|}_{{\mathrm{<figure-callout\; id="1"\; label="L"\; filenames="HDA00003765143700111.png,HDA00003765143700121.png"\; state="\{\{state\}\}">L</figure-callout>}}^1}}{{\left|\right|{\mathrm{\&Lambda;}}_r^{k-1}\left|\right|}_{L^1}}---\left(13\right)$

${\mathrm{\&epsiv;}}_v^k={\&Integral;}_{\mathrm{\&Omega;}}(1-g){\left|P^k\right|}_F+g{\left|P^k\right|}_{\mathrm{<figure-callout\; id="2"\; label="F"\; filenames="HDA00003765143700111.png,HDA00003765143700121.png"\; state="\{\{state\}\}">F</figure-callout>}}^2+\mathrm{\&eta;}{\&Integral;}_{\mathrm{\&Gamma;}}|s-n^{*}|---\left(14\right)$

Figure 12 (a) illustrates the figure of relative residual error (use equation (11) calculate) and iterations, and (figure in Figure 12 (a) has shown 10000 iteration (being log10000=4 on log scale) altogether, Figure 12 (b) illustrates the relative error (using equation (13) to calculate) of Lagrange multiplier and the figure of iterations, Figure 12 (c) illustrates the figure of numerical value energy value (use equation (14) calculate) and iterations, and this is all with respect to Fig. 2 (d) (top) with 18(b) for (bottom).Figure 12 (a) and axle (b) are all log scales, and, in Figure 12 (c), the y axle is log scale, and the x axle is the decimal scale.In Figure 12 (a), R1, R2 and R3 are respectively with respect in equation (12)

and in Figure 12 (b), L _r, L _c, and L _frespectively with respect in equation (13)

Figure in Figure 12 (a)-(c) has numerically shown that above-mentioned numerical algorithm restrains.The relative error of Lagrange multiplier has identical degree of convergence with relative residual error.And the numerical value energy reaches steady state (SS).Note, the result shown in the figure in this application is in the situation that be far smaller than 1000 iteration and obtain; In Table 1.In exemplary value experiment, we find, after meeting relative residual error ratio error range and being less than 0.001 condition, almost can not in reconstructed surface, detect any visible difference.

As shown in Figure 12 (a)-(c), the relative error of residual sum Lagrange multiplier has identical degree of convergence relatively, and the figure of numerical value energy value just reaches steady state (SS) under iteration seldom.Note the catch hell impact of value of parameter of the speed of layout convergence.Larger punishment parameter value can make the numerical value energy value reduce in iteration slower (, numerical value energy layout need to spend the longer time to restrain), and slower (that is, the relative error layout restrains the time of Hua Gengchang with relative residual error layout) that less punishment parameter value will make the relative error in Lagrange multiplier reduce in iteration with relative residual error.The preferred punishment parameter of adjusting, so that the relative error layout restrains sooner with relative residual error layout.

Be below the exemplary algorithm of replacing Method for minimization, as mentioned above, this replaces the equation (7) in Method for minimization resolving Algorithm 1.

algorithm 2: the replacement Method for minimization of solve an equation (7)

Step 1: initializing variable: ${\tilde{s}}^0=s^{k-1},{\tilde{n}}^0=n^{k-1}$ With ${\tilde{P}}^0=P^{k-1}.$

Step 2: for l=1 ..., L and fixing Lagrange multiplier

with

optionally solve following minimization problem:

${\tilde{s}}^l=\arg \min L_v(s,{\tilde{n}}^{l-1},{\tilde{P}}^{l-1};{\mathrm{\&lambda;}}_f,{\mathrm{\&lambda;}}_c,{\mathrm{\&Lambda;}}_r)---\left(15\right)$

${\tilde{n}}^l=\arg \min L_v({\tilde{s}}^l,n,{\tilde{P}}^{l-1};{\mathrm{\&lambda;}}_f,{\mathrm{\&lambda;}}_c,{\mathrm{\&Lambda;}}_r)---\left(16\right)$

${\tilde{P}}^l=\arg \min L_v(s^l,{\tilde{n}}^l,P;{\mathrm{\&lambda;}}_f,{\mathrm{\&lambda;}}_c,{\mathrm{\&Lambda;}}_r)---\left(17\right)$

By farthest reducing energy functional (18), (19) and (20) find out above-mentioned minimizer

with

${\mathrm{\&epsiv;}}_1\left(s\right)={\&Integral;}_{\mathrm{\&Omega;}}\frac{{\mathrm{<figure-callout\; id="2"\; label="c\; f"\; filenames="HDA00003765143700111.png,HDA00003765143700121.png"\; state="\{\{state\}\}">c</figure-callout>}}_f}{2}{|s-{\tilde{n}}^{l-1}|}^2+{\mathrm{\&lambda;}}_f\&CenterDot;s+\mathrm{\&eta;}{\&Integral;}_{\mathrm{\&Gamma;}}|s-n^{*}|---\left(18\right)$

${\mathrm{\&epsiv;}}_2\left(n\right)={\&Integral;}_{\mathrm{\&Omega;}}\frac{{\mathrm{<figure-callout\; id="2"\; label="c\; r"\; filenames="HDA00003765143700111.png,HDA00003765143700121.png"\; state="\{\{state\}\}">c</figure-callout>}}_r}{2}{|{\tilde{P}}^{l-1}-\&dtri;n|}_F^2-{\mathrm{\&Lambda;}}_r\&CenterDot;\&dtri;n+\frac{c_c}{2}{(\&dtri;\&times;n)}^2+{\mathrm{\&omega;}}_c{\mathrm{\&lambda;}}_c(\&dtri;\&times;n)$

$+\frac{{\mathrm{<figure-callout\; id="2"\; label="c\; f"\; filenames="HDA00003765143700111.png,HDA00003765143700121.png"\; state="\{\{state\}\}">c</figure-callout>}}_f}{2}{|{\tilde{s}}^l-n|}^2-{\mathrm{\&lambda;}}_f\&CenterDot;n---\left(19\right)$

${\mathrm{\&epsiv;}}_3\left(P\right)={\&Integral;}_{\mathrm{\&Omega;}}(1-g){\left|P\right|}_F+g{\left|P\right|}_{\mathrm{<figure-callout\; id="2"\; label="F"\; filenames="HDA00003765143700111.png,HDA00003765143700121.png"\; state="\{\{state\}\}">F</figure-callout>}}^2+\frac{{\mathrm{<figure-callout\; id="2"\; label="c\; r"\; filenames="HDA00003765143700111.png,HDA00003765143700121.png"\; state="\{\{state\}\}">c</figure-callout>}}_r}{2}{|P-\&dtri;{\tilde{n}}^l|}_{\mathrm{<figure-callout\; id="2"\; label="F"\; filenames="HDA00003765143700111.png,HDA00003765143700121.png"\; state="\{\{state\}\}">F</figure-callout>}}^2+{\mathrm{\&Lambda;}}_r\&CenterDot;P---\left(20\right)$

Below will describe and how carry out the more details that farthest reduce functional (18)-(20).

Step 3: after the L iteration, by arranging upgrade (s ^k, n ^k, P ^k).In an example, (that is, L=1), but this has enough realized results needed only to have carried out an iteration.

minimizing of functional (18)-(20)

Minimizing of functional (18) and (20) (equation (15) and (16)) can solve by the soft-threshold method [25] that comprises simple algorithm calculations, and minimizing of functional (19) (equation (17)) stops by the system of separating linear equation on whole territory Ω.Because the grid of the system of linear equation is uniformly, so available fast Fourier transform method is calculated to be these equations of original solution to hang down.

Can be at discrete calculation territory Ω=[1, N ₁] * [1, N ₂] variable s in augmentation Lagranian functional in upper defining equation (6), n, P, λ _f, λ _c, Λ _r.Figure 13 (a)-13 (b) illustrates a lattice system.In lattice system physically different position be used for estimating variate-value.

Figure 13 (a) illustrates the variable s in index augmentation Lagranian functional in one example, n, P, λ _f, λ _c, Λ _rrule.In this example, n, s and λ _fin first and second elements be defined in respectively square place and hollow dots o place, and p ₁₂, p ₂₁, Λ _r12and Λ _r21be defined in the solid dot place.Other variable-definition is at triangle △ place, wherein, and λ _c(i, j) is at triangle 1302 places, p ₁₁(i, j) and Λ _r11(i, j) is at triangle 1304 places, p ₂₂(i, j) and Λ _r22(i, j) is at triangle 11306 places.These index rules can be used in Euler-Lagrange equation (19) discrete.Figure 13 (b) illustrates and is being of a size of 5x4(, [1,5] x[1,4]) the discrete calculation territory in implement the example of the rule in Figure 13 (a).

The details of the example of functional (18)-(20) in the step 2 that at utmost reduces algorithm 2 below will be described.

minimize ε in equation (18) ₁ (s)

Indicate ε so ₁(s) functional in can be expressed as two parts, as shown in equation (21)-(23).

ε ₁(s)=ε _Ω\Γ(s)+ε _Γ(s) (21)

Wherein

${\mathrm{\&epsiv;}}_{\mathrm{\&Omega;}\backslash \mathrm{\&Gamma;}}\left(s\right)\&equiv;{\&Integral;}_{\mathrm{\&Omega;}\backslash \mathrm{\&Gamma;}}{\mathrm{\&lambda;}}_f\&CenterDot;s+\frac{{\mathrm{<figure-callout\; id="2"\; label="c\; f"\; filenames="HDA00003765143700111.png,HDA00003765143700121.png"\; state="\{\{state\}\}">c</figure-callout>}}_f}{2}{|s-n|}^2---\left(22\right)$

${\mathrm{\&epsiv;}}_{\mathrm{\&Gamma;}}\left(s\right)\&equiv;{\&Integral;}_{\mathrm{\&Gamma;}}\mathrm{\&eta;}|s-n^{*}|+{\mathrm{\&lambda;}}_f\&CenterDot;s+\frac{{\mathrm{<figure-callout\; id="2"\; label="c\; f"\; filenames="HDA00003765143700111.png,HDA00003765143700121.png"\; state="\{\{state\}\}">c</figure-callout>}}_f}{2}{|s-n^{*}|}^2---\left(23\right)$

Because inscape is the quadratic polynomial about s, so can be easy to obtain this first energy functional ε _{Ω Γ}(s) minimizer (minimizer).In order to obtain the second energy functional ε _Γ(s) minimizer in, be expressed as equation (24) again by equation (23).In equation (24), the C value is independent of and minimizes.

${\mathrm{\&epsiv;}}_{\mathrm{\&Gamma;}}\left(s\right)\&equiv;{\&Integral;}_{\mathrm{\&Gamma;}}\mathrm{\&eta;}|s-n^{*}|+\frac{{\mathrm{<figure-callout\; id="2"\; label="c\; f"\; filenames="HDA00003765143700111.png,HDA00003765143700121.png"\; state="\{\{state\}\}">c</figure-callout>}}_f}{2}{|s-n^{*}+\frac{{\mathrm{\&lambda;}}_f}{c_f}|}^2+C---\left(24\right)$

For each coordinate (i, j) ∈ Γ, all used soft-threshold method [32], and by under (25)-(26) that establish an equation obtain the minimizer in equation (15)

wherein $\mathrm{\&alpha;}=\max (\mathrm{0,1}-\frac{\mathrm{\&eta;}}{c_f\left|x_0\right|})$ With ${\mathrm{<figure-callout\; id="0"\; label="x"\; filenames="HDA00003765143700111.png,HDA00003765143700121.png"\; state="\{\{state\}\}">x</figure-callout>}}_0=n^{*}-\frac{{\mathrm{\&lambda;}}_f}{c_f}.$

$(i,j)\&NotElement;\mathrm{\&Gamma;}\&DoubleRightArrow;{\tilde{s}}^l(i,j)=n(i,j)-\frac{1}{c_f}{\mathrm{\&lambda;}}_f(i,j)---\left(25\right)$

$(i,j)\&Element;\mathrm{\&Gamma;}\&DoubleRightArrow;{\tilde{s}}^l(i,j)=n^{*}(i,j)+\mathrm{\&alpha;}(i,j)x_0(i,j)---\left(26\right)$

minimize ε in equation (19) ₂

For what fix with

euler-Lagrange equation (19) produces the system of the linear partial differential equation (PDE) as shown in equation (27), wherein, and n=(n ₁, n ₂) ^t,

with ▽ P=(▽ p ₁, ▽ p ₂) ^t, I is the 2x2 unit matrix, with $D_c={\&dtri;}^{\&perp;}{\&dtri;}^{{\&perp;}^T}.$

-(c _rD _r+c _cD _c-c _fI)n=▽·Λ _r-▽ ^⊥(ω _cλ _c)+λ _f-▽·P+c _fs (27)

Because D _rlaplce (Laplacian) operator, and D _roperate on each element of n, so its existence in equation (27) makes equation (27) become a coupled wave equation, and make n ^*on stroke, spread.Yet the differentiating operator in equation (27) (because the pressure of the irrotationality on fine and close vector field constraint exists) produces unnecessary power, with by n ^*be inserted on whole territory.

The standard finite difference scheme that there is index variables rule as shown in Figure 13 (a) by use, the n in equation (27) ₁discrete at square place, and the n in equation (27) ₂discrete at hollow dots o place.If periodic boundary condition is applied to all variablees, just can changes F with discrete fourier and realize that equation (27) is discrete.The shift operator that comprises the finite differential in equation (27) mainly means discrete convolution, and the discrete fourier of this discrete convolution is changed in frequency field as to multiply each other by component.

For discrete frequency, can form the linear equation system shown in figure (28).

$\left(\begin{array}{cc}{a}_{11}& a_{12}\\ a_{21}& a_{22}\end{array}\right)\left(\begin{array}{c}{\mathrm{Fn}}_1(u_i,u_j)\\ {\mathrm{Fn}}_2(u_i,u_j)\end{array}\right)=\left(\begin{array}{c}{f}_1(u_i,u_j)\\ f_2(u_i,u_j)\end{array}\right)---\left(28\right)$

Indicate

${\mathrm{\&zeta;}}_i=2{\sin }^2\frac{{\mathrm{<figure-callout\; id="2"\; label="v\; i"\; filenames="HDA00003765143700111.png,HDA00003765143700121.png"\; state="\{\{state\}\}">v</figure-callout>}}_i}{2}+\sqrt{-1}\sin v_i---\left(29\right)$

${\mathrm{\&zeta;}}_j=2{\sin }^2\frac{{\mathrm{<figure-callout\; id="2"\; label="v\; j"\; filenames="HDA00003765143700111.png,HDA00003765143700121.png"\; state="\{\{state\}\}">v</figure-callout>}}_j}{2}+\sqrt{-1}\sin v_j,---\left(30\right)$

Wherein, $v_i=\frac{2\mathrm{\&pi;}}{N_1}{u}_i(u_i=1,...,N_1)$ With $v_j=\frac{2\mathrm{\&pi;}}{N_2}{u}_j(u_j=1,...,N_2),$ Coefficient in the system of linear equation (28) is

$a_{11}=4c_r{\sin }^2\frac{{\mathrm{<figure-callout\; id="2"\; label="v\; i"\; filenames="HDA00003765143700111.png,HDA00003765143700121.png"\; state="\{\{state\}\}">v</figure-callout>}}_i}{2}+4(c_r+c_c){\sin }^2\frac{{\mathrm{<figure-callout\; id="2"\; label="v\; j"\; filenames="HDA00003765143700111.png,HDA00003765143700121.png"\; state="\{\{state\}\}">v</figure-callout>}}_j}{2}+c_f$

$a_{12}=c_c(-{\stackrel{\&OverBar;}{\mathrm{\&zeta;}}}_j){\mathrm{\&zeta;}}_i$ $a_{21}=c_c(-{\stackrel{\&OverBar;}{\mathrm{\&zeta;}}}_i){\mathrm{\&zeta;}}_j$

$a_{22}=4(c_r+c_c){\sin }^2\frac{{\mathrm{<figure-callout\; id="2"\; label="v\; i"\; filenames="HDA00003765143700111.png,HDA00003765143700121.png"\; state="\{\{state\}\}">v</figure-callout>}}_i}{2}+4c_r{\sin }^2\frac{{\mathrm{<figure-callout\; id="2"\; label="v\; j"\; filenames="HDA00003765143700111.png,HDA00003765143700121.png"\; state="\{\{state\}\}">v</figure-callout>}}_j}{2}+c_f---\left(31\right)$

And, the f on the right-hand side of equation (28) ₁and f ₂can express as follows:

${\mathrm{<figure-callout\; id="1"\; label="f"\; filenames="HDA00003765143700111.png,HDA00003765143700121.png"\; state="\{\{state\}\}">f</figure-callout>}}_1={\stackrel{\&OverBar;}{\mathrm{\&zeta;}}}_{j}F{\mathrm{\&alpha;}}_{12}^1+{\stackrel{\&OverBar;}{\mathrm{\&zeta;}}}_{i}F{\mathrm{\&alpha;}}_{11}^2-F{\mathrm{\&alpha;}}_1^3---\left(32\right)$

${\mathrm{<figure-callout\; id="2"\; label="f"\; filenames="HDA00003765143700111.png,HDA00003765143700121.png"\; state="\{\{state\}\}">f</figure-callout>}}_2={\stackrel{\&OverBar;}{\mathrm{\&zeta;}}}_{j}F{\mathrm{\&alpha;}}_{21}^1+{\stackrel{\&OverBar;}{\mathrm{\&zeta;}}}_{i}F{\mathrm{\&alpha;}}_{22}^2-F{\mathrm{\&alpha;}}_2^3---\left(33\right)$

Wherein

${\mathrm{\&alpha;}}_{\mathrm{mn}}^1\&equiv;{\mathrm{\&Lambda;}}_{\mathrm{rmn}}+c_r(p_{\mathrm{mn}}-p_{\mathrm{nm}})+{(-1)}^m{\mathrm{\&omega;}}_c{\mathrm{\&lambda;}}_c---\left(34\right)$

${\mathrm{\&alpha;}}_{\mathrm{mn}}^2\&equiv;{\mathrm{\&Lambda;}}_{\mathrm{rmn}}+c_r{p}_{\mathrm{mn}}---\left(35\right)$

${\mathrm{\&alpha;}}_{\mathrm{<figure-callout\; id="3"\; label="m"\; filenames="HDA00003765143700111.png,HDA00003765143700121.png"\; state="\{\{state\}\}">m</figure-callout>}}^3\&equiv;{\mathrm{\&lambda;}}_{\mathrm{fm}}+c_f{s}_m---\left(36\right)$

There is a N ₁n ₂the 2x2 system, and above-mentioned equation is for all discrete frequencies, and the determinant of the matrix of coefficients in above-mentioned equation is provided by equation (37), wherein

penalty parameter c _r, c _cand c _ffor on the occasion of, so this determinant is non-vanishing.For each frequency solution after the linear equation system, just use discrete fourier to reverse and bring acquisition

(4c _rβi _j+c _f)(4(c _r+c _c)βi _j+c _f) (37)

minimize ε in equation (20) ₃ (P)

Indicate

functional in equation (20) can remake following expression.In equation (38), the C value is independent of and minimizes.

${\mathrm{\&epsiv;}}_3\left(P\right)={\&Integral;}_{\mathrm{\&Omega;}}(1-g){\left|P\right|}_F+\frac{c_r+2\mathrm{<figure-callout\; id="2"\; label="g"\; filenames="HDA00003765143700111.png,HDA00003765143700121.png"\; state="\{\{state\}\}">g</figure-callout>}}{2}{|P-\frac{1}{c_r+2g}(c_r\&dtri;n-{\mathrm{\&Lambda;}}_r)|}_{\mathrm{<figure-callout\; id="2"\; label="F"\; filenames="HDA00003765143700111.png,HDA00003765143700121.png"\; state="\{\{state\}\}">F</figure-callout>}}^2+C---\left(38\right)$

Can use following method [8] to find out the closed form of minimizer at each point (i, j) ∈ Ω place:

$p(i,j)=\max \{\mathrm{0,1}-\frac{1-g(i,j)}{c_r{\left|W(i,j)\right|}_F}\}W(i,j)---\left(39\right)$

Wherein

$W=\frac{c_r}{c_r+2g}(c_r\&dtri;n-{\mathrm{\&lambda;}}_r)---\left(40\right)$

step 108: from fine and close vector field, rebuild the 3D surface

In step 108, with height map z=I (x ₁, x ₂) the 3D surface that is associated rebuilds from fine and close vector field n on computational fields Ω, fine and close vector field n obtains in step 106.This operation is carried out by placing height figure I, and the degree of tilt field of height map meets fine and close vector field n.

In one example, the 3D surface is to rebuild by the weighted combination of two methods, and two methods are TV rule method and H ¹rule method, the acutance of TV rule method for preserving reconstructed surface, H ¹the smoothness of rule method for preserving reconstructed surface.This operation can be based on as shown in equation (41) energy functional ε _h(I) minimize, this equation can be carried out TV rule method and H ¹the weighted combination of rule method.In equation (41), I ₀be illustrated in

upper required height value, h is weighting function (or indicator function), its arbitrary value between desirable 0 and 1.Similar to equation (2), equation (41) has shown the minimizing of Nonlinear Functional of height map, this Nonlinear Functional comprise a plurality of of the upper definition of territory Ω (surface rebuilds thereon) and the upper definition of territory Σ (presenting the territory about the prior imformation of height).About the prior imformation of height, in other words, elemental height information, can be determined by the user, makes modeling process similar to the deformation process that becomes another height map from a height map.And some form of elemental height information can comprise the vector on the part territory.In one example, if there is no the available prior imformation about height, will be made as 0 with regard to ξ so.And, can carry out weighting function h by the convolution of the 2D gaussian kernel with smoothing factor 2.

$\underset{I}{\min }\{{\mathrm{\&epsiv;}}_h\left(I\right)={\&Integral;}_{\mathrm{\&Omega;}}(1-h)|\&dtri;I-n|+{\&Integral;}_{\mathrm{\&Omega;}}h{|\&dtri;I-n|}^2+\mathrm{\&xi;}{\&Integral;}_{\mathrm{\&Sigma;}}|I-{\mathrm{<figure-callout\; id="0"\; label="I"\; filenames="HDA00003765143700111.png,HDA00003765143700121.png"\; state="\{\{state\}\}">I</figure-callout>}}_0\left|\right\}---\left(41\right)$

In equation (41), | ▽ I-n| means TV type norm (that is, the TV rule method), and | ▽ I-n| ²mean H ¹type norm (that is, H ¹rule method).TV type norm is preserved the acutance of reconstructed surface and is rebuild discontinuous point (for example, jump discontinuity) and not there will be surface deformation, and H ¹the type norm is by impelling the C of height map ⁰continuity is strengthened balance (because the ∫ of reconstructed surface _Ωh| ▽ I-n| ²generate Laplace operator in Euler-Lagrange equation).

Because jump and may suffer from excessive adjusting or regulate not enough problem in reconstructed surface, so, preferably not at whole territory Ω application H ¹type norm (that is, not whole territory Ω is set to h=1) and on the other hand, when h ≠ 1, used TV type norm to contribute to prevent from excessively regulating or regulating not enough problem.In one example, consider TV and H ¹the different-effect of type norm, just the classification based on stroke and be respectively each stroke and weighting function h(be set, weight) value.For example, for the great-jump-forward stroke, this value just is made as 0, and just is made as 1 for other strokes.In another example, the position of great-jump-forward stroke is known in advance, and the h value is just based on this prior imformation Lookup protocol.

minimizing in equation (4)

In one example, the minimization problem of equation (41) solves by the expansion Lagrangian method, as follows.Note, also have some other for the method for the energy function that minimizes equation (41), as discussed in list of references [39]-[42].Identical method also is applied to minimizing of energy function in equation (2).

The minimization problem of equation (41) at first is converted to constrained minimization problem, as shown in equation (42), wherein, q=▽ I and J=I.

$\underset{I}{\min }{\&Integral;}_{\mathrm{\&Omega;}}(1-h)|q-n|+{\&Integral;}_{\mathrm{\&Omega;}}h{|q-n|}^2+\mathrm{\&xi;}{\&Integral;}_{\mathrm{\&Sigma;}}|J-{\mathrm{<figure-callout\; id="0"\; label="I"\; filenames="HDA00003765143700111.png,HDA00003765143700121.png"\; state="\{\{state\}\}">I</figure-callout>}}_0|---\left(42\right)$

Then, minimization problem is converted to unconstrained problem, as shown in equation (43), and has found out the saddle point of the augmentation Lagranian functional in equation (43).In equation (43), d _rand d _fjust to punish parameter, μ _rand μ _fit is Lagrange multiplier.

$L_h(\&CenterDot;)\&equiv;{\&Integral;}_{\mathrm{\&Omega;}}(1-h)|q-n|+h{|q-n|}^2$

$+{\&Integral;}_{\mathrm{\&Omega;}}{\mathrm{\&mu;}}_r\&CenterDot;(q-\&dtri;I)+\frac{{\mathrm{<figure-callout\; id="2"\; label="d\; r"\; filenames="HDA00003765143700111.png,HDA00003765143700121.png"\; state="\{\{state\}\}">d</figure-callout>}}_r}{2}{\&Integral;}_{\mathrm{\&Omega;}}{|q-\&dtri;I|}^2$

$+{\&Integral;}_{\mathrm{\&Omega;}}{\mathrm{\&mu;}}_f(J-I)+\frac{{\mathrm{<figure-callout\; id="2"\; label="d\; f"\; filenames="HDA00003765143700111.png,HDA00003765143700121.png"\; state="\{\{state\}\}">d</figure-callout>}}_f}{2}{\&Integral;}_{\mathrm{\&Omega;}}{(J-I)}^2+\mathrm{\&xi;}{\&Integral;}_{\mathrm{\&Sigma;}}|I-{\mathrm{<figure-callout\; id="0"\; label="I"\; filenames="HDA00003765143700111.png,HDA00003765143700121.png"\; state="\{\{state\}\}">I</figure-callout>}}_0|---\left(43\right)$

Can find out the optimal condition of equation (43) by using the iterative algorithm similar with algorithm 2 to above-mentioned algorithm 1.The more details of this iterative algorithm will be described in list of references [11], and the content of list of references [11] is incorporated herein by reference.When ξ ≠ 0 or h on the Ω of territory ≠ 1, usually use this iterative algorithm.

If do not have available about the prior imformation of desired height and great-jump-forward stroke not, ξ=0 so, h is made as h=1, and equation (41) only comprises and means H ¹the item of type norm.In this case, Euler-Lagrange equation I is only a Poisson (Poisson) equation, as shown in equation (44).Therefore, equation (41) does not need to solve with iterative algorithm.On the contrary, minimizing of the energy functional in equation (41) can only be realized by the Poisson equation in solve an equation (44).In other words, the solution of the Poisson equation in equation (44) is a reconstruction height map, and the degree of tilt of this reconstruction height map meets vector field n(, and this is due to the irrotationality constraint of forcing on equation (2)).

▽·(▽I)=▽·n (44)

Because the vector field n inserted meets integrability condition, so, in the possibility of the Xie Zhongke of equation (44) elimination surface deformation.Therefore, if the degree of tilt of height map meets vector field n, the surface be associated with this height map rebuild so not there will be surface deformation.

step 110: on the 3D surface rebuild, carry out Local Edit

Step 110 is optional steps, for revise surface details on reconstructed surface.Because assessing the cost of step 110 is not high, so, can in step 110, carry out in real time and calculate.

The whole territory that need not again insert reconstructed surface in order on reconstructed surface, to revise surface details.On the contrary, in one example, at first user's sketch is revised by add at least another pen on sketch, and then, the place based on revising in user's sketch redefines computational fields.Then, use the computational fields redefined to carry out equation (2) and (41).

Figure 14 (a) and the surface that 14(b) illustrates respectively user's sketch and rebuild from user's sketch, and Figure 14 (c) and 14(d) illustrate respectively and there is the user's sketch (that is, thering is Local Edit) that adds stroke and there is the reconstructed surface corresponding to the stroke additional surface details of interpolation.

As shown in Figure 14 (c), added strokes in the local field 1402 of sketch, the placed around of the stroke in local field rectangular shaped rim 1404(manually or automatically).Then, make following modification corresponding to the part of the 1404 pairs of reconstructed surfaces of rectangular shaped rim in sketch.

For the interpolation stroke based in Figure 14 (c), the additional surface details is incorporated to reconstructed surface, therefore redefines computational fields.In one example, Ω redefines the territory into being surrounded by rectangular shaped rim 1404, and Γ redefines the stroke (comprising the stroke of new interpolation, the stroke existed before and zone 1406) of surrounding for rectangular shaped rim 1404.Zone 1406 comprises the zone that deducts the zone in border 1402 in rectangular shaped rim 1404.Then, along stroke Γ allocation vector, then use equation (2) to insert the vector of distribution to form fine and close vector field.Then, reconstructed surface is revised by using equation (41) to rebuild this part corresponding to the part of rectangular shaped rim in sketch.Come accounting equation (2) and (41) with the territory Ω redefined and stroke Γ.And, in equation (41), the elevation information that Σ can be made as on 1406, the Σ of zone can be used as I ₀.As shown in Figure 14 (d), be incorporated to the surface details added, but the geometric figure of reconstructed surface before not changing.At Figure 14 (a)-14(d) shown in example in, territory is of a size of the 64x64 pixel, what the Local Edit step was used is less than 1 second.Note, although Figure 14 illustrates single local field 1402, the two or more local fields of separating of definable in single sketch.

the contrast of method 100 and existing algorithm

With the existing systematic comparison based on WIMP, it is more directly perceived that using method 100 is rebuild the 3D surface, time of cost less and do not need a large amount of work.And, by the 3D surface of existing SBIM system reconstructing, such as, SKETCH[12], Teddy[13], SmoothSketch[14] and FiberMesh[15], usually all coarse or all level and smooth and be circle.In other words, by the 3D surface that has the SBIM system reconstructing now, usually do not comprise surface details, for example, the shape of folding line.And method 100 does not have these restrictions.

Method 100 equally also is better than LUMO[6] and ShapePalettes[7] in method.Vector insertion method in LUMO and ShapePalettes is linear, and the sparse vector that distributes of the reflection that reconstructed surface can not be strictly according to the facts.And on the other hand, owing to there being the irrotationality constraint, method 100 can reflect the vector be arranged in reconstructed surface faithfully.Although this also needs non-linear vector to insert, in method 100, used the quick solver based on Augmented Lagrange method [8] can carry out fast so that non-linear vector inserts.

In one example, an artist has described figure as shown in Figure 15 (a) shows with the form of sketch, and Figure 15 (e) illustrates the vector distributed along the stroke in sketch.Figure 15 (b) and figure (f), figure (c) and figure (g) and figure (d) and figure (h) illustrate respectively the reconstructed surface of fine and close vector field, use LOMO method, ShapePalettes method and method 100 acquisitions.As shown in Figure 15 (f) and figure (g), because the promotion of the linearity in LUMO and ShapePalettes method and the constraint of shortage irrotationality, so the quilt of the ripple on cloth " floating " in reconstructed surface.In the ShapePalettes method, because the irrotationality constraint does not meet the jump in jump discontinuity and surface normal, so the irrotationality constraint can cause the surface deformation in reconstructed surface.If the prior imformation of the known position about this discontinuous point, just may prevent from forming surface deformation when using appropriate weighting function.Yet this prior imformation is seldom known.On the other hand, because non-linear vector inserts the irrotationality constraint with reinforcement and has used TV rule method and Augmented Lagrange method, so the surface that using method 100 is rebuild clearly comprises the ripple on cloth.

In another example, by using prior art approach and method 100 from the sketch reconstructed surface of the cat as shown in Figure 16 (a).Figure 16 (b)-(d) illustrates respectively the surface of using LUMO method, ShapePalettes method and method 100 to rebuild.As shown in these figures, LUMO method and ShapePalette method can only be rebuild the 3D surface with limit shape, and can not preserve surface details.On the other hand, due to the irrotationality condition of using in method 100 and non-linear insertion, so the surface details in the reconstructed surface in Figure 16 (d) is more sharp-pointed, that is, preserve better.Note, although the LUMO method also can generate the reconstructed surface similar to Figure 16 (d) with the ShapePalettes method, also need a large amount of extra effort (for example, draw extra stroke and carry out extra vector setting).These extra steps can be not intuitively.

And single view modeling method [9,10] needs a large number of users interactive (and time) with constraint qualification, compares with most of this single view modelings [9,10], the vector assignment method in method 100 is directly perceived and maneuverable.Equally, processing sparse degree of tilt, preserve the acutance feature and prevent that the surface deformation in surface-gradient construction is convenient, [7], the algorithm in [9] and [10] is all inconvenient, the author in list of references [16] has pointed out the surface deformation of surface-gradient in building (list of references [16] used do not strengthen discrete gaussian kernel method that can be long-pending overcome this difficulty).And on the other hand, because method 100 has been used the TV rule method and has been forced the irrotationality condition with non-linear vector insertion, so method 100 can not faced identical difficulty.But method 100 provides a kind of very quick also effective method in order to solve the energy functional in non-linear vector insertion by Augmented Lagrange method.

Google SketchUp[17] be based on the 3D modeling software of sketch, it allows intuitively and the fast modeling of 3D surface.Yet, in order to use the program in SketchUp, the user needs manually to specify the height on 3D surface.By mouse drag, operate or keyboard input is carried out this operation.On the other hand, but the height on the 3D surface rebuild in automatic estimation method 100.And the system in SketchUp is not the single view modeling, but and this system using method 100 is carried out.And the process of using method 100 is also tended to consistent with the traditional design stage, because design phase in early days, in modeling and operation 3D plane, before the modeling of sketch, devisers are usually at 2D plane picture sketch.In addition, method 100 also can distribute to come the surface of modeling complexity with simple stroke and rapid vector, as shown in Figure 17 (a)-(c).On the other hand, creating the free type surface with SketchUp may be very difficult.

the advantage of embodiments of the invention

It is below the major advantage of method 100.

in method 100 in conjunction with the TV rule method andh ¹ rule method

Before, resurfacing is rebuild from the sparse information based on the energy minimization method, and this resurfacing is carried out by using visible surface to show [30], wherein, and Weighted H ¹the norm data are controlled the local smoothing method degree of reconstructed surface together with its degree of tilt with discrete fidelity.Yet, estimate that an appropriate weighting function is very difficult for this reason.Another is TV norm data [31] for the method for controlling local smoothness before.Great-jump-forward discontinuous point under TV norm data is easy in the situation that do not have weighting function to preserve, and the discontinuous point of surface slope can recover by TV norm data skewness.

In method 100, TV rule method and H ¹rule method combination under weighting scheme, in order to carry out the resurfacing based on stroke.By changing the texture that can easily control reconstructed surface in conjunction with weight used.This not only can make artists freely create various types of surface geometries, also help geometrically important structure (seeing Figure 10 and Figure 19) and be conducive to the preservation of surface details and can not cause surface deformation (therefore, reconstructed surface comprises sharp-pointed surface characteristics) of reconstruction.Because the use of TV rule method, so produce some above-mentioned advantages.The TV rule method not only is conducive to preserve the discontinuous point in vector field, also helps the discontinuous point in the degree of tilt of preserving reconstructed surface, and can not cause surface deformation.Therefore, the great-jump-forward discontinuous point can be kept in reconstructed surface and not have excessive adjusting or regulate not enough.

in method 100, use the non-linear vector with irrotationality condition to insert

In method 100, the stroke in user's sketch is distributed sparse vector field, by carrying out non-linear vector to insert, from the sparse vector field of distributing, obtains fine and close vector field.This insertion is that the irrotationality condition by forcing on fine and close vector field is carried out.

Non-linear vector in method 100 inserts the reflection strictly according to the facts that has realized that vector arranges, and the insertion vector produced in method 100 almost all meets and can amass (irrotationality) condition in whole territory.This has preserved better the surface details of reconstructed surface and can not cause surface deformation, has therefore realized more desired structures (seeing Figure 11) corresponding to given stroke and vector setting.

directly perceived and the hommization of method 100

Method 100 can create reconstructed surface as shown in Figure 2 (d) shows, and method 100 only arranges to create this reconstructed surface by service regeulations stroke and acquiescence initialization vector.In other words, method 100 can be used according to the stroke of artistical typical case's drawing style and carry out reconstructed surface.Artists draw stroke with the style of oneself usually, and stroke type can consuming timely be set or revise vector to create how required surface.Therefore, the directly perceived and hommization of method 100.

Initialization vector can be automatically assigned on the stroke in user's sketch, and initialization vector is based on for example to be given tacit consent to the initialization vector setting and is assigned on stroke.In one example, the vector of distribution is identical with the vector magnitude of stroke tangent line quadrature and distribution.Automatically by vector assignment on the stroke in user's sketch than manual allocation vector (for example, based on as [7] in the 3D list of references modeling carried out carry out allocation vector) have more advantage, because it has reduced user's workload, therefore make method 100 hommization more.

Yet the vector automatically distributed still can be regulated in order to form more detailed surface configuration by the user.In one example, the user can regulate amplitude and the direction of allocation vector, the user based on its imagination to required surface by revising with keyboard or mouse.This numerical value than input table presentation surface height is more directly perceived.

can create dissimilar by using different stroke types and/or weighting function g surface geometry.

In order to rebuild more complicated surface, but the various stroke types of user's choice for use create different how much key structures, such as, carinate, paddy shape, great-jump-forward, raised water and tilting stroke.In method 100, according to required surface to be formed, stroke is classified.Different stroke types is conducive to the user and creates desired structure in reconstructed surface.These structures can intuitively imagine from corresponding 2D stroke (seeing Figure 18 and 19), and therefore, artists can create can be from the surface of 2D stroke visual perception.The geometric figure of significant vector based on required surface and being assigned on various types of strokes geometrically.The vector of these distribution can be that the surface normal vector is in the projection had on the 2D plane of user's sketch.Method 100 can successfully by these, sparse allocation vector be transformed on the desired structure of reconstructed surface.

Also can form more complicated surface by the weighting function g regulated in equation (2).For example, comprise the surface of discontinuous point in order to rebuild degree of tilt, sketch can be divided into to different texture regions, the user only need determine which texture region should comprise discontinuous point.This is for example, unlike existing method (, the algorithm in list of references [16]) so dull tediously long, for existing method, must show the accurate location of degree of tilt discontinuous point in territory.

Figure 19 (a) illustrates user's sketch of monkey.In Figure 19 (a), user's sketch is divided into two parts: the texture part on the right and the left side do not add texture part.Texture part on the right arranges g=0, and the texture part that do not add of on the left side arranges g=1, and this has just produced the reconstructed surface in Figure 19 (b).As shown in Figure 19 (b), this reconstructed surface comprises different geometries, and the geometry on the right is sharp-pointed, and the geometry on the left side is level and smooth.

Figure 19 (c)-19(d) illustrate with various stroke types and rebuild the surface with degree of tilt or great-jump-forward discontinuous point; Figure 19 (c) illustrates user's sketch of the monkey with different strokes, and Figure 19 (d) illustrates the surface rebuild from Figure 19 (c).As shown in Figure 19 (d), reconstructed surface has been preserved discontinuous point and without any distortion, and reconstructed surface clearly comprises the feature required according to the stroke type in user's sketch.

the speed of method 100 and low assessing the cost

With the algorithm in for example list of references [16], compare, method 100 needs stroke still less creates the reconstructed surface of extra fine quality.And, can carry out Local Edit in method 100, carry out Local Edit by adding several surface details to previous reconstructed surface, need in whole territory, not insert.Yet it only need to redefine territory by the location revision based in user's sketch.

And, in method 100, use Augmented Lagrange method to be conducive to overcome the difficulty run in solving constrained minimization problem.Even in method 100, vector insertion and height map are rebuild and all carried out non-linear process, but result still can be by realizing with quick and effective numerical value solver, the numerical evaluation device is based on Lagrangian method [32].The speed that this has reduced to rebuild required computing time of the modeling process on 3D surface and has improved modeling process.In method 100, can low assess the cost and high computing velocity under by with Augmented Lagrange method the GPU based on CUDA, calculating to carry out non-linear insertion.In one example, by using tall and handsome Da Kezhuo (nVidia Quadro) FX1800, the territory that size is 512x512 pixel and 256x256 pixel needs respectively to be less than the computing time of 15 seconds and 5 seconds.Low computing velocity can be depending on user's sketch.

Table 1 illustrates iterations and the computing time (Intel (R) Xeon (R) CPU E5520@2.27GHz, NVIDIA Quadro FX1800) for creating reconstructed surface shown in some above-mentioned figure of execution.

If sketch comprises great-jump-forward, raised water or tilting stroke, in order to rebuild the surface in table 1 diagram, just carry out twice insertion.Carrying out insertion for the first time is in order from regular, carinate and paddy shape stroke, to rebuild the 3D surface, and carrying out insertion for the second time is in order after inserting for the first time, to create great-jump-forward, raised water and tilting structure on reconstructed surface.This has preserved the geometric figure of great-jump-forward, raised water or tilting structure periphery, as shown in Fig. 2 (d), 10,18 and 19.In table 1, " 1 " means that the result in this figure is to insert for the first time rear acquisition, and " 2 " mean that the result in this figure is to insert for the second time rear acquisition.

If occur ξ=0 and h=1(on Ω, user's sketch does not comprise the great-jump-forward stroke), just do not need to solve an equation (41) with iterative algorithm, this is because only by equation (41), obtain height map.In table 1, the broken number of selecting "-" in " iterations " hurdle in step 108 means, in order to obtain the result in figure, does not need to carry out iteration with solve an equation (41).The broken number of selecting "-" in step 108 in " time " hurdle means not perform step 108.In order to rebuild the surface in table 1 diagram, if in user's sketch without the great-jump-forward stroke, so by following parameter η=100, c _r=10, c _c=1 and c _f=1 for equation (6).If the great-jump-forward stroke is arranged in user's sketch, so by following parameter ξ=0 and μ _r=0.01 for reconstruction procedures, that is, and and in equation (43).

And, for the statistics in reckoner 1, the error range ε in algorithm 1 is made as 0.001 in step 106, if be made as 0.05(in step 108, need iterative algorithm), in table 1, the error range of all data all arranges like this.Use table 1 is contrasted, the consistance of the statistics that kept assessing the cost.Yet, be noted that the computing time needed in order to reduce, can select some larger error ranges to rebuild some surfaces.

As shown in table 1, in above-mentioned table, reconstructed surface does not need iteration.And the quality of reconstructed surface is quite high, and can obtain at short notice.Therefore, can draw a conclusion thus: can fast and hang down and be calculated to be local implementation method 100.

Table 1

a plurality of application of method 100

Method 100 not only can be rebuild static 3D surface.It can also create the animation sequence by a series of static 3D surface compositions by changing gradually the vector be assigned on user's sketch.For example, can rebuild the 3D surface from user's sketch iteratively for a plurality of frameworks of animation sequence and create, wherein, the vector distributed along the stroke in user's sketch in each iteration is all different.Therefore, method 100 can be used on making of cartoon short-movie montage in animation tool.Figure 20 (a)-(d) illustrates a plurality of frameworks of selecting the example animation sequence formed from using method 100.The amplitude of the vector that change distributes along the stroke that forms first surface, when each vector magnitude changes, with regard to reconstructed surface, animation sequence generates like this.Each rebuilds the 3D surface is all the framework of animation sequence.By not only changing the amplitude of the vector distributed, also change its direction, can form more complicated animation sequence like this.

And method 100 also can be used in other application, for example, 2D Cartoon shading [35], [36] and [37] or geometric error revision [38] etc.The surface modeling software that also but using method 100 is developed based on sketch, this software can be used in various designs application, for based on user's sketch, also rebuilding exactly easily, fast complicated 3D surface.

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Claims (20)

1. the method for the reconstruction of three-dimensional surface, territory from two-dimentional sketch reconstruction of three-dimensional surface, described two-dimentional sketch comprises at least one stroke, described at least one stroke is assigned vector field, said method comprising the steps of:

Insert described vector field to form fine and close vector field; And

Use described fine and close vector field to rebuild described surface;

Wherein, inserting described vector field comprises: the Nonlinear Functional that farthest reduces described vector field, there is the irrotationality constraint on described fine and close vector field, the Nonlinear Functional of described vector field comprises the item defined on a plurality of and the described at least one stroke defined on territory, wherein, use the augmented Lagrangian method farthest to reduce the Nonlinear Functional of described vector field.

2. method according to claim 1, wherein, insert described vector field and further comprise: use TV rule method and H ¹the weighted array of rule method, the acutance of described TV rule method for preserving reconstructed surface, described H ¹the smoothness of rule method for preserving reconstructed surface.

3. method according to claim 2, wherein, insert described vector field and further comprise:

Required texture based on surface to be reconstructed and be described sketch installation weight;

By come the described TV rule method of weighting and described H by described weight ¹the combination of rule method and insert described vector field.

4. method according to claim 3, wherein, for the step of described sketch installation weight further comprises following sub-step:

Described sketch is sprayed to the color be associated with variable alpha value; And

Change described alpha value in order to be described sketch installation weight.

5. method according to claim 2, wherein, described method further comprises the step that described sketch is divided into to a plurality of parts, and, to each part in described a plurality of parts, insert described vector field and further comprise following sub-step:

Required texture based on surface to be reconstructed and be that described part arranges weight; And

The weight that is described part setting by use is come the described TV rule method of weighting and described H ¹the combination of rule method and insert described vector field in described part.

6. according to the described method of any one in the claims, wherein, rebuild described surface and comprise: use TV rule method and H ¹the weighted array of rule method, the acutance of described TV rule method for preserving reconstructed surface, described H ¹the smoothness of rule method for preserving reconstructed surface.

7. according to the described method of any one in the claims, wherein, rebuilding described surface comprises: use the augmented Lagrangian method farthest to reduce the Nonlinear Functional of height map, the Nonlinear Functional of described height map comprises item defined on a plurality of and another territory defined on territory, in described another territory, has about prior imformation highly.

8. the method for the reconstruction of three-dimensional surface, territory from two-dimentional sketch reconstruction of three-dimensional surface, described two-dimentional sketch comprises at least one stroke, said method comprising the steps of:

Distribute a plurality of vectors in order to form vector field along described at least one stroke;

By rebuild described three-dimensional surface in described territory according to the described method of any one in the claims.

9. method according to claim 8, wherein, before along described at least one stroke, distributing a plurality of vectors, described method further comprises the step of described at least one stroke being classified according to the surface geometry for the treatment of to rebuild from described at least one stroke.

10. method according to claim 9, the step of wherein, described at least one stroke being classified further comprises: by described at least one stroke be divided into regular stroke, shape ridge stroke, paddy shape stroke, great-jump-forward stroke, tilting stroke and raised water stroke one of them.

11. the described method of any one according to Claim 8-10 wherein, distributes a plurality of vectors further to comprise following sub-step along described at least one stroke:

Distribute a plurality of initialization vectors along described at least one stroke, wherein, the tangent line quadrature of described a plurality of initialization vectors and described at least one stroke;

Revise described a plurality of initialization vector to form a plurality of vectors that distribute along described at least one stroke.

12. method according to claim 11, wherein, the sub-step of revising described a plurality of initialization vectors further comprises: revise the amplitude of described a plurality of initialization vectors and direction one of them or all revise.

13. the described method of any one according to Claim 8-12, wherein, described at least one stroke shows will form in described reconstructed surface once jumps, the direction of the described a plurality of vectors that distribute along described at least one stroke and direction and the amplitude that amplitude means respectively jump to be formed.

14. the described method of any one according to Claim 8-12, wherein, described at least one stroke shows will form an oblique line in described reconstructed surface, the direction of the described a plurality of vectors that distribute along described at least one stroke and direction and the degree of tilt that amplitude means respectively oblique line to be formed.

15. method according to claim 9, and depend on claim 3, wherein, for the step of described sketch installation weight comprises: for described at least one stroke arranges weight, before the step of inserting described vector field, described method is further comprising the steps:

Classification based on described at least one stroke and revise the weight of described at least one stroke.

16., according to the described method of any one in the claims, described method is further comprising the steps:

Revise described sketch by add at least another stroke to described sketch;

Described at least another stroke placed around rectangular shaped rim in described sketch;

Revise part corresponding with rectangular shaped rim described in sketch in described reconstructed surface.

17. method according to claim 16, wherein, described at least one stroke further comprises: be arranged in a stroke of described rectangular shaped rim, revise part corresponding with the described rectangular shaped rim in sketch in described reconstructed surface and further comprise following sub-step:

Described at least another stroke and a described stroke in described rectangular shaped rim are distributed a plurality of vectors;

Insert described a plurality of vector to form another fine and close vector field;

By from described another fine and close vector field, rebuilding the described part that described reconstructed surface is revised described reconstructed surface.

18. one kind from comprising that the two-dimentional sketch of at least one stroke generates the method for animation sequence, said method comprising the steps of:

Utilize the described method of any one according to Claim 8-17, by from two-dimentional sketch, creating a plurality of frameworks for described animation sequence in the reconstruction of three-dimensional surface iteratively, wherein, a plurality of vectors that distribute along described at least one stroke are all different in each iteration.

19. a computer system, it has for carrying out the processor according to the described method of claim 1-18 any one.

20. a computer program, such as, tangible data memory device, can be read and be comprised by the exercisable instruction of the processor of computer system by computing machine, and described instruction is carried out according to the described method of any one in claim 1-18 described processor.

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